We present an adaptive finite element method for vector-valued semilinear elliptic PDEs that is rate-optimal with respect to computational cost, i.e., computation time. To ensure linear complexity of the individual building blocks of the adaptive algorithm, we adaptively linearize the underlying semilinear PDE and solve the arising symmetric positive definite system by means of a norm-contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid method. To deal with the local Lipschitz continuity of the problem, we prove that the norm of all computed iterates of the proposed adaptive iteratively linearized finite element method (AILFEM) are uniformly bounded. Owing to an equibalance of discretization, linearization, and algebraic solver errors, the algorithm guarantees optimal convergences rates with respect to the number of degrees of freedom, computational cost, and computation time.